Lipschitz Operators Research Articles

Operator and commutator moduli of continuity for normal operators

We study in this paper properties of functions of perturbed normal operators and develop earlier results obtained in Aleksandrov, Peller, Potapov and Sukochev [‘Functions of normal operators under perturbations’, Adv. Math. 226 (2011) 5216–5251.]. We study operator Lipschitz and commutator Lipschitz functions on closed subsets of the plane. For such functions we introduce the notions of the operator modulus of continuity and of various commutator moduli of continuity. Our estimates lead to estimates of the norms of quasicommutators f(N1)R−Rf(N2) in terms of ‖N1R−RN2‖, where N1 and N2 are normal operator and R is a bounded linear operator. In particular, we show that if 0 < α < 1 and f is a Hölder function of order α, then, for normal operators N1 and N2, ‖ f ( N 1 ) R − R f ( N 2 ) ‖ ⩽ const ( 1 − α ) − 2 ‖ f ‖ Λ α ‖ N 1 R − R N 2 ‖ α ‖ R ‖ 1 − α . In the last section, we obtain lower estimates for constants in operator Hölder estimates.

Functions of perturbed tuples of self-adjoint operators

We generalize earlier results of Aleksandrov and Peller (2010) [2,3], Aleksandrov et al. (2011) [6], Peller (1985) [13], Peller (1990) [14] to the case of functions of n-tuples of commuting self-adjoint operators. In particular, we prove that if a function f belongs to the Besov space B∞,11(Rn), then f is operator Lipschitz and we show that if f satisfies a Hölder condition of order α, then ‖f(A1,…,An)−f(B1,…,Bn)‖⩽constmax1⩽j⩽n‖Aj−Bj‖α for all n-tuples of commuting self-adjoint operators (A1,…,An) and (B1,…,Bn). We also consider the case of arbitrary moduli of continuity and the case when the operators Aj−Bj belong to the Schatten–von Neumann class Sp.

Functions of perturbed dissipative operators

We generalize our results of \cite{AP2} and \cite{AP3} to the case of maximal dissipative operators. We obtain sharp conditions on a function analytic in the upper half-plane to be operator Lipschitz. We also show that a Holder function of order $\a$, $0<\a<1$, that is analytic in the upper half-plane must be operator Holder of order $\a$. Then we generalize these results to higher order operator differences. We obtain sharp conditions for the existence of operator derivatives and express operator derivatives in terms of multiple operator integrals with respect to semi-spectral measures. Finally, we obtain sharp estimates in the case of perturbations of Schatten-von Neumann class $\bS_p$ and obtain analogs of all the results for commutators and quasicommutators. Note that the proofs in the case of dissipative operators are considerably more complicated than the proofs of the corresponding results for self-adjoint operators, unitary operators, and contractions that were obtained earlier in \cite{AP2}, \cite{AP3}, and \cite{Pe6}.

On generalized Newton method for solving operator inclusions

In this paper, we study the existence and uniqueness theorem for solving the generalized operator equation of the form F(x) + G(x) + T(x) ? 0, where F is a Fr?chet differentiable operator, G is a maximal monotone operator and T is a Lipschitzian operator defined on an open convex subset of a Hilbert space. Our results are improvements upon corresponding results of Uko [Generalized equations and the generalized Newton method, Math. Programming 73 (1996) 251-268].

The numerical radius of Lipschitz operators on Banach spaces

We study the numerical radius of Lipschitz operators on Banach spaces. We give its basic properties. Our main result is a characterization of finite-dimensional real Banach spaces with Lipschitz numerical index&nbsp;$1$. We also explicitly compute the Lip

A product formula for semigroups of Lipschitz operators associated with abstract quasilinear evolution equations

AbstractThe notion of semigroups of Lipschitz operators associated with abstract quasilinear evolution equations is introduced and a product formula for such semigroups is established. The product formula obtained in the paper is applied to the solvability of the Cauchy problem for a first order quasilinear system through a finite difference scheme of the Lax‐Friedrichs type.

Strong convergence theorem for amenable semigroups of nonexpansive mappings and variational inequalities

In this paper, using strongly monotone and lipschitzian operator, we introduce a general iterative process for finding a common fixed point of a semigroup of nonexpansive mappings, with respect to strongly left regular sequence of means defined on an appropriate space of bounded real-valued functions of the semigroups and the set of solutions of variational inequality for β-inverse strongly monotone mapping in a real Hilbert space. Under suitable conditions, we prove the strong convergence theorem for approximating a common element of the above two sets.

Primal-Dual Splitting Algorithm for Solving Inclusions with Mixtures of Composite, Lipschitzian, and Parallel-Sum Type Monotone Operators

We propose a primal-dual splitting algorithm for solving monotone inclusions involving a mixture of sums, linear compositions, and parallel sums of set-valued and Lipschitzian operators. An important feature of the algorithm is that the Lipschitzian operators present in the formulation can be processed individually via explicit steps, while the set-valued operators are processed individually via their resolvents. In addition, the algorithm is highly parallel in that most of its steps can be executed simultaneously. This work brings together and notably extends various types of structured monotone inclusion problems and their solution methods. The application to convex minimization problems is given special attention.

On a wave equation with supercritical interior and boundary sources and damping terms

This article addresses nonlinear wave equations with supercritical interior and boundary sources, and subject to interior and boundary damping. The presence of a nonlinear boundary source alone is known to pose a significant difficulty since the linear Neumann problem for the wave equation is not, in general, well-posed in the finite-energy space H1(Ω) × L2(∂Ω) with boundary data in L2 due to the failure of the uniform Lopatinskii condition. Further challenges stem from the fact that both sources are non-dissipative and are not locally Lipschitz operators from H1(Ω) into L2(Ω), or L2(∂Ω). With some restrictions on the parameters in the model and with careful analysis involving the Nehari Manifold, we obtain global existence of a unique weak solution, and establish exponential and algebraic uniform decay rates of the finite energy (depending on the behavior of the dissipation terms). Moreover, we prove a blow up result for weak solutions with nonnegative initial energy.

A product formula for semigroups of Lipschitz operators associated with semilinear evolution equations of parabolic type

A product formula for semigroups of Lipschitz operators associated with semilinear evolution equations of parabolic type is discussed under a new type of stability condition which admits “error term”. The result obtained here is applied to showing the convergence of approximate solutions constructed by a fractional step method to the solution of the complex Ginzburg–Landau equation.

Functions of normal operators under perturbations

In Peller (1980) [27], Peller (1985) [28], Aleksandrov and Peller (2009) [2], Aleksandrov and Peller (2010) [3], and Aleksandrov and Peller (2010) [4] sharp estimates for f(A)−f(B) were obtained for self-adjoint operators A and B and for various classes of functions f on the real line R. In this paper we extend those results to the case of functions of normal operators. We show that if a function f belongs to the Hölder class Λα(R2), 0<α<1, of functions of two variables, and N1 and N2 are normal operators, then ‖f(N1)−f(N2)‖⩽const‖f‖Λα‖N1−N2‖α. We obtain a more general result for functions in the space Λω(R2)={f:|f(ζ1)−f(ζ2)|⩽constω(|ζ1−ζ2|)} for an arbitrary modulus of continuity ω. We prove that if f belongs to the Besov class B∞11(R2), then it is operator Lipschitz, i.e., ‖f(N1)−f(N2)‖⩽const‖f‖B∞11‖N1−N2‖. We also study properties of f(N1)−f(N2) in the case when f∈Λα(R2) and N1−N2 belongs to the Schatten–von Neumann class Sp.

Functions of perturbed normal operators

In Peller (1985, 1990) [10,11], Aleksandrov and Peller (2009, 2010, 2010) [1–3] sharp estimates for f ( A ) − f ( B ) were obtained for self-adjoint operators A and B and for various classes of functions f on the real line R . In this Note we extend those results to the case of functions of normal operators. We show that if f belongs to the Hölder class Λ α ( R 2 ) , 0 < α < 1 , of functions of two variables, and N 1 and N 2 are normal operators, then ‖ f ( N 1 ) − f ( N 2 ) ‖ ⩽ const ‖ f ‖ Λ α ‖ N 1 − N 2 ‖ α . We obtain a more general result for functions in the space Λ ω ( R 2 ) = { f : | f ( ζ 1 ) − f ( ζ 2 ) | ⩽ const ω ( | ζ 1 − ζ 2 | ) } for an arbitrary modulus of continuity ω. We prove that if f belongs to the Besov class B ∞ 1 1 ( R 2 ) , then it is operator Lipschitz, i.e., ‖ f ( N 1 ) − f ( N 2 ) ‖ ⩽ const ‖ f ‖ B ∞ 1 1 ‖ N 1 − N 2 ‖ . We also study properties of f ( N 1 ) − f ( N 2 ) in the case when f ∈ Λ α ( R 2 ) and N 1 − N 2 belongs to the Schatten–von Neumann class S p .

Interpolation of nonlinear operators in weighted L p -spaces

We show that Peetre’s classical interpolation theorem in weighted L p -spaces is carried over to some classes of nonlinear operators containing in particular the Lipschitz operators and operators close to them in the properties satisfying less restrictive conditions than Lipschitz in each of the spaces of a Banach pair.

Operator Hölder–Zygmund functions

It is well known that a Lipschitz function on the real line does not have to be operator Lipschitz. We show that the situation changes dramatically if we pass to Hölder classes. Namely, we prove that if f belongs to the Hölder class Λ α ( R ) with 0 < α < 1 , then ‖ f ( A ) − f ( B ) ‖ ⩽ const ‖ A − B ‖ α for arbitrary self-adjoint operators A and B. We prove a similar result for functions f in the Zygmund class Λ 1 ( R ) : for arbitrary self-adjoint operators A and K we have ‖ f ( A − K ) − 2 f ( A ) + f ( A + K ) ‖ ⩽ const ‖ K ‖ . We also obtain analogs of this result for all Hölder–Zygmund classes Λ α ( R ) , α > 0 . Then we find a sharp estimate for ‖ f ( A ) − f ( B ) ‖ for functions f of class Λ ω = def { f : ω f ( δ ) ⩽ const ω ( δ ) } for an arbitrary modulus of continuity ω. In particular, we study moduli of continuity, for which ‖ f ( A ) − f ( B ) ‖ ⩽ const ω ( ‖ A − B ‖ ) for self-adjoint A and B, and for an arbitrary function f in Λ ω . We obtain similar estimates for commutators f ( A ) Q − Q f ( A ) and quasicommutators f ( A ) Q − Q f ( B ) . Finally, we estimate the norms of finite differences ∑ j = 0 m ( − 1 ) m − j ( m j ) f ( A + j K ) for f in the class Λ ω , m that is defined in terms of finite differences and a modulus continuity ω of order m. We also obtain similar results for unitary operators and for contractions.

CONVERGENCE THEOREMS FOR THE ZEROS OF A FINITE FAMILY OF GENERALIZED ACCRETIVE OPERATORS

A strong convergence theorem for the common zero for a finite family of Generalized Lipschitz operators in a uniformly smooth Banach space is proved when atleast one of the operator is Generalized \(\Phi\)- accretive, using a new iteration formula. Similar result for Generalized Lipschitz and Generalized \(\Phi\)- pseudocontractive map is also proved. Our result extends the convergence results of Chidume [4] to a finite family improving many other results.

A Lipschitz semigroup approach to two-dimensional Navier–Stokes equations

Motivated by the notion of integral solutions in the sense of Bénilan, a dissipative condition combined with a tangential condition is introduced. A generation problem of semigroups of Lipschitz operators is discussed under such a dissipative condition and the result is applied to the initial–boundary value problem for the Navier–Stokes equation in two-dimensional space.

Modular metric spaces, II: Application to superposition operators

Applying the theory of modular metric spaces developed in the first part of this paper [V.V. Chistyakov (2009) [1]] we define a metric semigroup and an abstract convex cone of functions of finite generalized variation in the approach of Schramm [M. Schramm, Functions of Φ -bounded variation and Riemann–Stieltjes integration, Trans. Amer. Math. Soc. 287 (1) (1985) 49–63], which are significantly larger as compared to the spaces of bounded variation in the sense of Jordan, Wiener–Young and Waterman. We present a complete description of generators of Lipschitz continuous, bounded and some other classes of superposition Nemytskii operators mapping in these semigroups and cones, which extends recent results by Matkowski and Miś [J. Matkowski, J. Miś, On a characterization of Lipschitzian operators of substitution in the space BV 〈 a , b 〉 , Math. Nachr. 117 (1984) 155–159], Maligranda and Orlicz [L. Maligranda, W. Orlicz, On some properties of functions of generalized variation, Monatsh. Math 104 (1987) 53–65], Zawadzka [G. Zawadzka, On Lipschitzian operators of substitution in the space of set-valued functions of bounded variation, Rad. Mat. 6 (1990) 279–293] and Chistyakov [V.V. Chistyakov, Mappings of generalized variation and composition operators, J. Math. Sci. (New York) 110 (2) (2002) 2455–2466, V.V. Chistyakov, Lipschitzian Nemytskii operators in the cones of mappings of bounded Wiener φ -variation, Folia Math. 11 (1) (2004) 15–39].

The Solvability of a New System of Nonlinear Variational-Like Inclusions

We introduce and study a new system of nonlinear variational-like inclusions involving - -maximal monotone operators, strongly monotone operators, -strongly monotone operators, relaxed monotone operators, cocoercive operators, -relaxed cocoercive operators, - -relaxed cocoercive operators and relaxed Lipschitz operators in Hilbert spaces. By using the resolvent operator technique associated with - -maximal monotone operators and Banach contraction principle, we demonstrate the existence and uniqueness of solution for the system of nonlinear variational-like inclusions. The results presented in the paper improve and extend some known results in the literature.

A hybrid steepest-descent method for variational inequalities in Hilbert spaces

A Mann-type hybrid steepest-descent method for solving the variational inequality ⟨F(u*), v − u*⟩ ≥ 0, v ∈ C is proposed, where F is a Lipschitzian and strong monotone operator in a real Hilbert space H and C is the intersection of the fixed point sets of finitely many non-expansive mappings in H. This method combines the well-known Mann's fixed point method with the hybrid steepest-descent method. Strong convergence theorems for this method are established, which extend and improve certain corresponding results in recent literature, for instance, Yamada (The hybrid steepest-descent method for variational inequality problems over the intersection of the fixed-point sets of nonexpansive mappings, in Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications, D. Butnariu, Y. Censor, and S. Reich, eds., North-Holland, Amsterdam, Holland, 2001, pp. 473–504), Xu and Kim (Convergence of hybrid steepest-descent methods for variational inequalities, J. Optim. Theor. Appl. 119 (2003), pp. 185–201), and Zeng, Wong and Yao (Convergence analysis of modified hybrid steepest-descent methods with variable parameters for variational inequalities, J. Optim. Theor. Appl. 132 (2007), pp. 51–69).

Operator-based non-linear vibration control system design of a flexible arm with piezoelectric actuator

Most real plants must deal with some non-linear characteristics. For the non-linear plants, design of non-linear control system using operator-based robust right coprime factorisation is considered. The considered non-linear control system applications show the desired results. In this paper, operator-based non-linear vibration control of a flexible arm experimental system with uncertainties using robust right coprime factorisation is considered. In general, the vibrations by the characteristic vibration mode include from one to infinite order vibration modes and appear on the arm vibration. It is difficult to consider all vibration modes. Consequently, based on the concept of Lipschitz operators and robust right coprime factorisation condition, robust stability of the closed-loop system with unconsidered vibration modes as uncertainties is guaranteed, and desired vibration control performance actuated by piezoelectric element is ensured. Finally, simulation and experimental results are presented to support the effectiveness of the proposed design.